Final answer:
To find the length of the string when under tension, use the formula T = √(Fg/μL), where T is tension, Fg is the weight of the string, μ is the linear mass density, and L is the length of the string. Rearrange the equation to solve for L and substitute the given values to find the length of the string.
Step-by-step explanation:
To find the length of the string, we can use the formula T = √(Fg/μL), where T is tension, Fg is the weight of the string, μ is the linear mass density, and L is the length of the string. In this case, the tension is 20 N and the weight of the string is 25 g. We can convert the weight to Newtons by multiplying by the acceleration due to gravity, which is approximately 9.8 m/s². So the weight of the string is (25 g)(9.8 m/s²) = 0.245 N. Plugging these values into the formula, we have 20 N = √(0.245 N/μL).
One way to solve for the length is to square both sides of the equation to eliminate the square root: (20 N)² = (0.245 N/μL)². Simplifying, we get 400 N² = 0.060 N²/(μ²L²). Rearranging the equation, we have (μ²L²)(400 N²) = 0.060 N². Dividing both sides by 0.060 N², we get (μ²L²) = (400 N²)/(0.060 N²). Canceling out the N², we have (μ²L²) = 6666.67. Taking the square root of both sides, we have L = √(6666.67/μ²). Plugging in the value of μ = 0.006 kg/m, we can calculate:
L = √(6666.67/(0.006 kg/m)²) = √(6666.67/(0.000006 kg²/m²)) = √(1.11112 × 10¹⁵ m⁴/kg²). So the length of the string is approximately 3.33 × 10⁷ m.